A fuzzy interval multiobjective mixed integer programming approach for the optimal planning of solid waste management systems

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 89 (1997) 35 60

A fuzzy interval multiobjective mixed integer programming approach for the optimal planning of solid waste management systems N i - B i n C h a n g * , Y . L . C h e n , S.F. W a n g Department of Environmental Engineering, National Cheng-Kung University Tainan, Taiwan, R.O.C. Received July 1995; revised February 1996

Abstract

Various deterministic mathematical programming models were developed to evaluate single objective or multiple objectives planning alternatives for municipal solid waste management. The common objective of minimizing the present value of overall management cost/benefit was extended to deal explicitly with environmental considerations, such as air pollution, traffic flow limitation, and leachate and noise impacts. But uncertainty plays an important role in the search for sustainable solid waste management strategies. This paper proposes a new approach a fuzzy interval multiobjective mixed integer programming (FIMOMIP) model for the evaluation of management strategies for solid waste management in a metropolitan region. In particular, it demonstrates how uncertain messages can be quantified by specific membership functions and combined through the use of interval numbers in a multiobjective analytical framework. @ 1997 Elsevier Science B.V.

Keywords." Mathematical programming; Fuzzy sets; Management

1. Introduction

Multiple conflicting objectives generally characterize the current solid waste management systems. The goals of environmental quality protection and the needs of economic planning for solid waste management systems are not easy to be reconciled in decision making. Many studies for solid waste management were made of mathematical programming models. Specifically, the deterministic mathematical programming models, including linear programming (LP), mixed integer programming (MIP), dynamic programming (DP), and multiobjective programming, have been developed and applied for planning solid * Corresponding author 0165-0114/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PllS0165-0114(96)00086-3

waste management systems. However, it is recognized that the deterministic optimization techniques are not sufficient to model such a complex problem because of the inherent uncertainties in the decision maker's preference and the variations of parameter values in the solid waste management systems. Considering the limitations of the available data base and the model formalism, the fuzzy sets theory and interval programming techniques are identified as the prevailing approaches to handle such vagueness of the fuzzy goals and the uncertainties involved in the related parameter values simultaneously. The 'fuzzy interval multiobjective mixed integer programming' (FIMOMIP) model and its solution procedure are thus developed as a new analytical tool to deal with the locational problems of planning

36

Ni-Bin Chart9 et al./ Fuzz)' Sets and Systems 89 (1997) 35 60

regional solid waste management systems in this paper. It specifically shows how the interval messages related to the input parameter values and the fuzzy goals pertaining to the decision maker's aspiration levels are communicated into the multiobjective optimization processes. It is believed that such an analytical scheme would create a set of more flexible optimal planning alternatives. A case study for the evaluation of long-term management strategies for solid waste management in Taiwan was demonstrated. In addition to those essential functions in a solid waste management system, such as optimal waste stream allocation, siting, resources recovery, and tipping fees evaluation, environmental goals, consisting of the traffic congestion limitation, air pollution control, and noise impacts, are specifically regulated through the optimization process. By considering the vagueness existing in both environmental and economic parameters in systems analysis, interactions among the uncertain effects of waste generation, source reduction, recycling, collection, transfer, processing, and disposal are emphasized. The FIMOMIP model may then generate a set of total solution regarding waste recycling, facilities siting, and system operation. It unambiguously shows that the incorporation of interval and fuzzy types of messages as part of the decision making information not only reconciles the potential conflict between environmental and economic goals in the long run but also generates a set of flexible solid waste management policies.

2. Literature review

Various deterministic mathematical programming models have been used for planning solid waste management systems. The spectrum of these deterministic modelling techniques include linear programming (LP), mixed integer programming (MIP), dynamic programming (DP), and multiobjective programming. For example, Hsieh and Ho [21] and Lund et al. [37] discussed the optimization of solid waste disposal and recycling systems by linear programming techniques. But locational models, by using mixed integer programming (MIP) techniques, mostly have been used in real world applications [1, 1720, 28, 32, 34, 38, 44, 48]. The efforts in combining the environmental impacts, such as air pollution, leachate

impacts, noise control, and traffic congestion, as a set of risk constraints in an economic-oriented locational model were established by Chang et al. [3 6, l 1]. Perlack and Willis [42] further developed the analysis of multiobjective decision making in waste disposal planning. Chang and Wang [7] applied the compromise programming technique to ease the potential conflict during siting landfills, incinerators, and transfer stations in a growing metropolitan region. But uncertainty usually plays an important role in planning solid waste management problems. The random character governing sold waste generation, the estimation errors in parameter values, and the vagueness of planning objectives and constraints are all possible sources of uncertainty. In general, conventional mathematical programming approaches in dealing with system uncertainties can be classified by the following three methods: (1) stochastic programming approach application of probability theory; (2) interval or grey programming approach application of interval analysis; (3) fuzzy programming approach - application of fuzzy sets theory. Stochastic programming requires large size of data for the identification of the probability distribution. The focus of fuzzy sets theory is placed upon its non-statistical characteristics in nature that refers to the absence of sharp boundaries in the information. A subjective continuous membership function is usually used for the description of such type of information. However, the interval or grey programming approach emphasizes the vagueness of its intrinsic characteristics in the information during parameter estimation. Based on an interval analysis, with limited samples, the boundary of a parameter can be temporarily decided. Therefore, fuzzy sets and interval analysis can be used to supplement the interpretation of different types of uncertainties for real-world systems. In this decade, the fuzzy sets theory and interval programming technique have received wide attention in the field of planning solid waste management systems. For example, Koo et al. [33] accomplished the siting planning of regional hazardous waste treatment center using fuzzy multiobjective programming algorithm in Korea. Huang et al. [22-25] developed the grey linear programming (GLP), grey fuzzy linear programming (GFLP), grey fuzzy dynamic programming (GFDP), and grey integer programming (GIP) approaches in dealing with a hypothetical solid waste management problem in Canada. Recently, Chang et al. [9, 10]

37

Ni-Bin Chang et al./Fuzzy Sets and Systems 89 (1997) 35 60

applied fuzzy goal programming in dealing with several specific issues in the integrated solid waste management systems in Taiwan. Therefore, the fuzzy sets described by the membership functions and the grey parameters presented by a closed interval have been widely applied to supplement the use of traditional probability theory. In many environmental management issues, it is found that the parameters corresponding to the environmental or economical factors may encounter the grey type of message, while the imprecise objectives governed by the decision makers are much more adequately described by the prespecified fuzzy membership functions associated with their grey boundaries. This sort of membership function is unique and it is actually equivalent to the condition in which the membership function itself is fuzzy. Thus, the imprecise information involved in the decision making for the solid waste management can be further formulated by both fuzzy and interval expressions in a multiobjective analytical framework. This approach should result in more flexible solid waste management policies regarding the sustainable development in a metropolitan region.

3. The development of fuzzy interval multiobjective mixed integer programming model 3.1. Basic structure o f f u z z y and f u z z y interval multiobjective p r o g r a m m i n g

In the conventional multicriteria decision analysis, a deterministic multiobjective programming model is usually formulated to maximize and/or to minimize several objectives simultaneously, subject to a constraint set with 'greater than or equal to' and/or 'less than or equal to' relationships, as shown below. The equality constraints may be expressed as a combination of both types of inequality constraints. Bolditalic symbols in the following mathematical expressions represent the matrix or vector form for a set of variables or parameters in the context. Min

J}=C/X

V i = 1. . . . . m

(1)

Max

fj=CjX

Vj=m+l

(2)

..... m+n

subject to

AkX<~Bk

Vk = 1. . . . . p

(3)

AtX>~Bt

Vl = p +

(4)

1. . . . . p + q

(5)

x~>o

In the case of solid waste management modeling, sufficient information is frequently not available to assess their probability distributions. When human judgement is influential, fuzzy description is recognized as a better tool to define the preference level of decision makers. Hence, part or all of the above parameters in the matrices A, B and Ccould be expressed as fuzzy numbers, which might provide one of the flexibility in the formulation of such uncertainty and result in the fuzzy multiobjective programming model. Various types of membership functions can be used to support the fuzzy analytical framework although the fuzzy description is hypothetical and membership values are subjective. Nevertheless, the advantages of the inclusion of such linguistic expressions and the adoption of the following soft computing techniques would present a new realization of many natural and human systems. Since the goal of the fuzzy mathematical programming is to satisfy the fuzzy objective and constraints, a decision in a fuzzy environment is thus defined as the intersection of those membership functions corresponding to fuzzy objective and constraints [36,49]. In Eqs. ( 1 ) - ( 5 ) , if {Pc,,Pa2 . . . . . #a,,+,} and {Pc,,#c2,#c3 . . . . . #C,+q} are denoted as the membership functions for the fuzzy goals { G l , G2 ..... Gin+n} and fuzzy constraints { C l , C 2 . . . . . Cp+q}, respectively, in a decision space X, all the membership functions of Gm+n and Cp+q may then be combined to form a decision D, which is a fuzzy set resulting from the intersection of all related Gm+, and Cp+q, as shown below: D = Gl • G2 N . . • f-I Gm+n f-I Cl f'l C2 f-1. •. N Cp+q.

(6) The max-min convolution in decision making requires [49]: Max #D = Max min{#a,,#a2 . . . . . Pa ..... X

X

#c, , #c2 , Pc3 . . . . . #cp+q }.

(7)

Ni-Bin Chan9 et al./Fuzz)' Sets and Systems 89 (1997) 35 60

38

Thus, to build a fuzzy multiobjective programming model, the decision makers or analysts may express their aspiration levels, f l and f 2 , in advance that he or she wants to achieve for the values of the corresponding objective functions to be minimized and maximized, respectively, as well as each of the constraint modelled as a fuzzy set by a specific membership function. The fuzzy multiobjective programming, modified from Eqs. ( 1 ) - ( 5 ) , becomes:

(a) d e f i m t i o n of a n o n - i n c r e a s i n g linear m e m b e r s h i p timction:

bt i(EXi) -

1

1 (EXi fli) 8i 0

(8)

CjX ~ f 2 ,

(9)

AkX £Bk,

(10)

AzX~Bt,

(11)

X~>O,

(12)

where ' ~ ' and '~<' denote the fuzzified version of ' ~>' and ' ~<', which have the linguistic interpretation 'approximately larger than or equal to' and 'approximately smaller than or equal to', respectively. According to the direction of the inequalities, the above fuzzy multiobjective programming in Eqs. ( 8 ) - ( 1 2 ) can be summarized as:

EX £ R,

(13)

FX~S,

(14)

X~>O,

(15)

if EXi2 fli+ 8i

fli+ ~i

IP- EXi

(b) definition of a n o n - d e c r e a s i n g linear m e m b e r s h i p function: 1

if FXj> t~j

1 - (f2j-rxj) ~j 0

~tj(FXj) t

if f2j-~< FXj< f2j if FXj_< f2j-~j

-- -- - -

f2j -~j

f'2j

~ FXj

Fig. I. The expressions for fuzzy membership functions. can be determined by a conventional pay-offtable, as it is used for solving a deterministic multiobjective programming model [ 13, 46, 47]. Further, if introducing the variable of aspiration levels (i.e., the intermediate control variables) #i and p j, which represent of decision maker's preferences for different types of objectives, the max-rain convolution can be simplified as: Max #D = Max min {pi(x),ktj(x)}

where

X

(16) F=

if fli< EXi< fli+

~i(EXi)

fli

C~X £ f l ,

if EXi -
Cj

and

S=

(17)

In general, the non-increasing and non-decreasing linear membership functions are frequently used for the inequalities with 'less than or equal to' and 'greater than or equal to' relationships, respectively, as shown in Fig. 1. Using the membership functions defined in Fig. 1, we may assume that the membership values are linearly decreasing over the 'tolerance interval' 6i and linearly increasing over the 'tolerance interval' 6j. The unique information regarding the tolerances 6i and 6j

X

Vi,j.

(20)

Therefore, according to such a definition of a fuzzy decision in above, the membership function of the decision set, kiD(X) can be obtained by the max-min convolution based on Eqs. ( 8 ) - ( 11 ). Such an operation is actually an analogy to the non-fuzzy environment as the selection of activities simultaneously satisfy both the objective and constraints. If a unique aspiration level u is applied, the final formulation of a fuzzy multiobjective linear programming (FMOLP) model becomes: (21)

Max subject to

EX,. ~ f l j + (1 - #)6i

Vi

(22)

FXj >~f2j + tz6j

Vj

(23)

Ni-Bin Chang et al./Fuzzy Sets and Systems 89 (1997) 35 60

0~
(24)

xs>~0

Vj

(25)

Xs E X

Vs

(26)

It has been verified that the use of FMOLP consists of several merits compared to the traditional deterministic multiobjective programming [12,35]. At least, such an approach may further reduce computation time in search of the optimal solutions. However, the grey information, which can only be described as a closed interval rather than a prespecified membership function, has been frequently encountered in real world systems. In applications, it is found that the grey uncertainties involving in the input parameter values can propagate through the optimization analysis, which may further perturb the accuracy of the optimal solution and decrease the possibility in implementation. The conventional fuzzy programming approach is thus found unable to communicate such type of uncertain input information directly into the optimization processes and solutions. Therefore, the techniques of fuzzy programming and interval programming would be better combined together in the multiobjective programming models to present a more flexible mathematical structure. The following emphasis will be placed upon how to combine the fuzzy and interval programming models in a multiobjective analytical framework. In retrospect, interval analysis was first developed by Moore in 1979 in the US [39]. Later on, the grey systems theory, based on the concept of interval analysis, was introduced by Deng in 1984 in China [ 14 16], in which all systems are divided into three categories, including the white, grey, and black parts. While the white part shows completely certain and clear messages in a system, the black part has totally unknown characteristics. Hence, the messages released in the grey part is in between. In reality, fuzzy information is one type of grey message that the membership function can be exactly identified. Even so, if the fuzzy membership function itself is fuzzy, the description of the 'shadow' of the membership value involves the knowledge of interval analysis. Therefore, an interval or grey number, denoted as ®(a) in this analysis, may be defined as a closed interval with upper and lower limits [@(a), ~(a)]. Such an illustrative method supplements the expression of system uncertainties by

39

the conventional probability theory and fuzzy sets theory whenever the probability density and membership functions cannot be fully identified. In a FIMOMIP model, the left-hand side coefficients and the right-hand side stipulations in the constraints as well as the tolerance interval of aspiration levels in the objective function are defined as interval numbers, since some of those parameters in the systems frequently lack knowledge regarding how to specify the specific fuzzy membership functions. Hence, the use of interval programming techniques to improve the FMOLP model may present at least four contributions: (1) grey uncertainties embedded in the model parameters can be directly reflected and communicated into the optimization processes; (2) the variation or vagueness of the decision maker's aspiration level or preference level (i.e., the intermediate control variables) in the FMOLP model can further be narrowed down and thereby generate a more confident solution set for policy decision making; (3) regardless of the orientations of the decision maker's aspiration level (i.e., maximization or minimization of specific targets), each objective or goal may have its own independent membership function associated with its own grey message such that those objectives can be combined together and traded off in the multicriteria decision making process; and (4) the FIMOMIP configuration would automatically generate the most favorable optimal solution by a set of closed intervals, including the decision maker's aspiration levels and all the decision variables. It is not necessary to search for the satisfactory solution in a set of noninferior solutions by distance-based criteria, as required by the conventional solution procedure of the deterministic compromise programming model [ 13, 46, 47]. In the proposed FIMOMIP model, the conventional distinction between objectives and constraints no longer applies. The problem with multiple fuzzy objectives and grey parameters in the constraints is to find the optimal fuzzy decision D in a manner similar to the case in the traditional fuzzy environment. Two different groups of fuzzy objectives may be separately formulated in which a non-decreasing and a non-increasing linear membership functions with their corresponding mobile or grey tolerance interval are usually assumed for those objectives to be maximized and minimized, respectively. Based on the understanding that the fuzzy membership function itself

40

Ni-Bin Chan9 et aL / Fuzzy Sets and Systems 89 (1997) 35-60

is fuzzy, the expression of such a mobile tolerance interval must correspond to the grey fuzzy implications: 'the greater the better under an upper bound of the grey tolerance interval' or 'the smaller the better under a lower bound of the grey tolerance interval'. In Eq. (27), the grey intermediate control variable ®(c0 represents the degree of uncertainties associated with those inequalities with 'smaller than or equal to' and 'greater than or equal to' relationships in the fuzzy interval objective functions. Max-min convolution is then applied through the intersection of the degree of aspiration level of the corresponding objectives and the degree of approximation level of the corresponding inequality constraints. Thus the fuzzy objectives associated with their grey message can be arranged as a set of goal constraints, as shown in the following first and second constraint sets in Eqs. (28) and (29), respectively. The definitional grey fuzzy constraints are still retained in the decision space for the max-min convolution, as shown in Eqs. (30) and (31). Furthermore, in the systems analysis for environmental management, if a constraint is prepared for the fuzzy expression of 'the level of environmental resources consumption is substantially less than a given grey limitation', it is adequate to be described by a non-increasing linear membership function for the possible illustration of the fuzzy availability of the grey environmental resources. Such a constraint would not hold if we choose a non-decreasing linear membership function for a 'less than or equal to' inequality. On the other hand, for example, if the constraint is prepared to delineate the fuzzy expression of 'the minimum specific land development for sustainable yield of food product in a river basin or a reservoir watershed has to be substantially greater than a grey limitation', it is better to be described by a nondecreasing linear membership function for the possible illustration of the utilization of land resources. Similarly, such a constraint would not hold if we choose a non-increasing linear membership function for a 'greater than or equal to' inequality. However, both non-decreasing and non-increasing linear membership functions can be combined for the illustration of the equality constraint since the selection of the type of membership function is much more dependent on the actual physical meaning. Eqs. (30) and (31 ) fulfill the above mathematical thinking in applications.

Overall, 3i, 3j,3k, and 3l in Eqs. (34) and (35) are the tolerance intervals associated with each corresponding linear membership function in this analysis. The configuration of those non-increasing or nondecreasing linear membership functions with mobile tolerance intervals (i.e., the uncertainty of the fuzzy membership function itself) can be referred to as the superimposed image of two similar types of membership functions, based on the patterns in Fig. 1. This sort of description is consistent with the observable human hesitation frequently existing in the decision making process. The FIMOMIP model is thus formulated as:

max ® (~)

(27)

subject to: (1) constraints for the objectives to be minimized: ® ( q ) ®(x)<.o_(fi) + (1 - ® ( ~ ) ) [ ~ ( f )

- o_(f,)],

i = 1. . . . . m;

(28)

(2) constraints for the objectives to be maximized: ® ( c j ) ®(x)~> o(36) + ® ( ~ ) [ ~ ( £ ) j=m+l

- o(J))],

..... m + n ;

(29)

(3) constraint for the 'less than or equal to' grey fuzzy relationship:

®(A~) @(X) ~Q(ak)+(1 - @(cO)[~(Bk)-~(Bk)], k = 1. . . . . p;

(30)

(4) constraint for the 'greater than or equal to' grey fuzzy relationship: ®(AD ®(X)~>@(Bl) + ®(e)[@(Bt) - @(Bt)], 1 =p+l

(31)

..... p+q;

(5) membership constraint: 0~< ®(~)~< 1;

(32)

(6) non-negativity constraint: ®(xs)>~0, @(Xs)E@,

s = 1. . . . . r;

(33)

Ni-Bin Chan9 et al./ Fuzzy Sets and Systems 89 (1997) 35-60

41

® ( s ~) = [®(xT), ® ( x ~ ) . . . . . ®(x~*)],

(53)

(34)

®(x~') = [ Q ( x , ) , ~ ( x s ) ] ,

(54)

(35)

3.2. M e t h o d o f solution

® ( Q ) = {®(ci),®(c2) ..... ®(Cm)},

(36)

®(C~) = {®(Cm+~),®(Cm+2)..... ®(Cm+,,)},

(37)

®(X) = {®(x! ), ®(x2) . . . . . ®(xr)},

(38)

where C~i = @ ( j ~ ) -- @ ( f ) ,

6k = ~ ( B k )

(~j = @ ( £ ' )

-- @ ( D ) ,

- @_(B~), ,~ = ~ ( B ~ ) - @_(BI),

s = 1 . . . . . r.

®(Bk) = {®(bk)},

k = 1. . . . . p,

(41)

In the conventional deterministic multiobjective programming, such as compromise programming [46,47], unlimited satisfactory solutions are frequently examined for several planning altematives using the distance-based techniques. However, to obtain the optimal solution in the FIMOMIP, defined by Eqs. ( 2 7 ) - ( 3 3 ) , only two submodels are required to be solved independently and the combination o f both sets of optimal solutions from the two submodels would automatically constitute a set of fuzzy interval optimal solution, as described in Eqs. ( 5 0 ) - ( 5 4 ) . Those two submodels are:

® ( B / ) = {®(b/)},

l=p+l

(42)

max

® ( f ) = {®(f/)},

i = 1. . . . . m,

(43)

subject to:

@(fj)={®(J:)},

j=m+

(44)

® ( A k ) = {®(aij)},

i = 1. . . . . p, j = t . . . . . r,

(39) @(AI) = {®(aq)}, i = p + 1. . . . . p + q ,

(40)

j = 1. . . . . r,

..... p+q,

1. . . . . m + n .

~(~)

(55)

(1) constraints for the objectives to be minimized: 2 ( C l i )@(X) + @(C2i )@(X)

For grey vectors ® ( f ) and ®(B), and grey matrix ®(A), we have:

~®(f)

+ (1 - ~ ( ~ ) ) [ ~ ( f , ) - _@(~)],

i = 1. . . . . m;

®(a/j) = [ ~ ( a u ) , ~ ( a q ) ],

(56)

(2) constraints for the objective to be maximized: i = 1. . . . . p + q ,

j=

(45)

1. . . . . r,

@( C l j )@_@(X ) + @( C 2 j )@@(X )

®(bk) = {~(bk),~(bk)], ®(bt) = [@(bl),~(bl)],

k :

1. . . . . p,

(46)

j = m + 1. . . . . m + n;

l = p+ 1,...,p+q,

(47)

® ( f , ) = [O__(j5),#(:5)],

i=

1. . . . . m,

®(f:) = [__O(k),#(f:)],

j=m+l

(48)

(49) The optimal solutions for Eqs. ( 2 7 ) - ( 3 3 ) will be:

®(A.)

= [£(~.),~(X.)],

®(~.)

= [£(k.),~(D.)],

j=m+l

..... m+n,

(50) i = 1 . . . . . m,

(57)

(3) constraints for the 'less than or equal to' grey fuzzy relationship:

~(&)_@(x)

..... m+n.

®(~*) = [ o ( ~ * ) , ~ ( ~ * ) ] ,

~>_e(/)) + _ e ( ~ ) [ ~ ( / ) ) - _o(/))],

(51)

~<~(Bk) + (1 -- ~(c~))[~(Bk) -- ~(Bk)], k = 1 . . . . . p;

(58)

(4) constraint for the 'greater than or equal to' grey fuzzy relationship:

~(&)@_(x)

~>®(B/) + ~(c~) [ ~ ( B l ) - @(Bt)], (52)

l=p+l,...,p+q;

(59)

Ni-Bin Chan(] et al./Fuzzy Sets and Systems 89 (1997) 35 60

42

(5) membership constraint: 0~<®(~)~< 1;

(60)

(6) non-negativity constraint: @(Xs)>~O,@(Xs) E ~ X ,

s = 1. . . . . r;

(61)

and max

Q(ct)

(62)

subject to: ( 1) constraints for the objectives to be minimized:

o _ ( c l ~ ) ~ ( x ) + ~(c2~ ) V ( x ) < ® ( f ) + (1 - o _ ( ~ ) ) [ ~ ( f ) - _o(f)], i = 1. . . . . m;

(63)

(2) constraints for the objectives to be maximized: ~ ( C l j ) ~ ( X ) + O_(CZj)~(X)

~>~ ( f i ) + V ( ~ ) [ ~ ( £ ) - 0 ( ~ ) ] , j=m+

1. . . . . m + n ;

(64)

(3) constraints for the 'less than or equal to' grey fuzzy relationship:

_o(& )®(x) ~ ( B k ) + ( 1 - Q ( ~ ) ) [~(Bk ) - ~(Bk)],

k = 1. . . . . p;

(65)

(4) constraints for the 'greater than or equal to' grey fuzzy relationship:

_o(At)~(x) > ~ ( B I ) + ~ ( ~ ) [ ~ ( B I ) - ~(BI)], l=p+l

..... p+q;

(66)

(5) membership constraint: 0 ~<~(c~) < 1;

(67)

(6) non-negativity constraint: -~(Xs)>-O,~(xs) E ~ X ,

s = 1. . . . . r.

(68)

In the above submodels, C I i and Clj represent the terms with positive coefficients, while C2j and C2j stand for the terms with negative coefficients in those objective functions to be maximized or minimized. The reason for having such a distinction of positive and negative signs in those objective coefficients is simply to achieve the goals of maximization and minimization by the correct selections of the upper or lower bounds in the grey environment. The upper and lower bounds of those decision variables would then be identified in the grey fuzzy definitional constraints by the inherent inequality structure in each submodel. To build the grey fuzzy membership function, a payofftable, established by solving each GLP model stepwise, is required to find out the most attainable bounds for each objective when overall objectives are considered simultaneously. In each run, two submodels associated with each GLP model have to be further solved sequentially in order to accomplish each pair of attainable bounds within such a pay-off table [22, 23], and then the completed set of those attainable bounds can be used as the input parameters in the formulation of the final FIMOMIP model. After the establishment of the generation of optimal solutions in each GLP model, the fuzzy interval optimal outputs of the FIMOMIP model can then be determined through the use of those submodels in Eqs. (55 )-(68). However, an important step in the FIMOMIP solution procedure have to be emphasized in this analysis. In order to ensure that the locational pattern would be consistent in both sets of optimal solutions obtained from the two submodels of FIMOMIP analysis, as defined in Eqs. (55) (68), they must be solved sequentially. Consequently, the submodel 1, defined by Eqs. (55)-(61), should be solved first to determine the waste flow and site selection patterns as well as the upper bound of fuzzy interval optimal solution. Then, by imposing the conditionality constraints corresponding to the selected waste management pattern into submodel 2, the solution procedure of submodel 2, defined by Eqs. (55)-(68), is then performed to further search for the lower bound of the fuzzy interval optimal solution. The flowchart of the entire solution procedure of the FIMOMIP model is shown in Fig. 2. In such a solution procedure, a concept of grey degree is introduced before the final acceptance of a set of fuzzy interval optimal solution. As defined by Huang et al. [22, 23], the grey degree of a

43

Ni-Bin Chang et al./ Fuzz); Sets and Systems 89 (1997) 35-60

start

4. F I M O M I P model formulation for planning solid waste management system

)

I build the deterministic multiobjective programming model

investigate the grey intervals for those coefficients and stipulations and then modify the deterministic multiobjective programming model

solve each GLP submode[ and create the

I

pay-off table to determine the upper and / lower bounds of each objective function value[

build the FIMOMIPmodeland solve two submodels sequentially

find out fuzzy interval optimal solution

no

[

+

T =

~

Z,(®(c,)

-

®(B,))

t=l

I determine the best alternative I

,he end

q(®(sjk,))

Minimize

,~ yes

(

In the following model formulation, only part of the environmental and economic parameters or variables are expressed as interval numbers. Many technical parameters, such as design or expansion capacity of treatment or disposal facilities as well as those environmental standards, should not be considered as uncertain parameters or variables. The definition of all major variables are listed in the Appendix. Four fuzzy objectives, consisting of economics, noise control, air pollution control, and traffic congestion limitation, are considered in this analytical framework. The objective function for cost minimization is formulated for calculating the discounted cash flow of all quantifiable system benefits and costs over time. Discounted factors are equivalent to such an economic adjustment and provide the net system value for decision making. Hence, the real discounted factor is defined simultaneously by the inflation rate ( f ) and the nominal interest rate (r), which is denoted as fit(= [(1 + f ) / ( 1 + r)]t-1). The expression of the objective function is:

)

The aggregate cost

(®(Ct))

(69)

consist of:

• total transportation cost =

~'~

[®(CTjkt)

(j,k )El, j:fik

Fig. 2. The solution procedure of FIMOMIP model.

®(Sjkt)]

(70)

• total construction cost grey number is equivalent to its grey interval divided by its mid-value within the upper and lower bounds. It is expected that the grey degree generated from fuzzy interval outputs should be smaller than that of the interval solutions obtained in each single run of the GLP model. Overall, an important point in solving the FIMOMIP models is that the upper and lower bounds of objective function values obtained from a single run of each GLP model may not be the final upper and lower bounds of the corresponding objective function values in the FIMOMIP analysis because of the perturbation of the other objectives in the multicriteria decision space.

=

~

[ ® C q t DCk, + ®Fkt Ykt]

(71)

kc(J\Jt)

• total operating cost

= kc( J\J,~ UK\KI ) [®co~, ~ ®(sjk,)1 L (j,k )Cll

(72)

• total expansion cost =

~'~

[®CEkt TEXPkt]

(73)

kG(J\J] UK\KI )

• total recycling cost = ~ iGR

®(TRit) ®(CRit)

(74)

44

Ni-Bin Chan 9 et al./ Fuzzy Sets and Systems 89 (1997) 35-60

The aggregate benefit (®(BD) considered here are:

of standard garbage trucks needed (i.e., ®(Sjkt)+ Pt). The expression for the objective function is:

• total resource recovery income at the facilities =

4- ~

~

~

iER kE(MUK4UJ4) (j,k)cI~

[®(Pikt)®(Tikt) ®(Sjkt)] (75)

• total household recycling income = +

~

~ [®(IRijt)

iE( Ji UKI ) jER

®(~ijt) ®(G/x)]

Minimize

r

=Z

In the expression, set subtraction is represented by a backslash (\). The total transportation costs are expressed as linearly proportional to unit waste loading, and the average operating cost is assumed to be a constant. As usual, a fixed charge structure is employed in the formulation of total construction cost for the purpose of site selection. However, the facilities expansion cost does not have a fixed charge term, and only the variable cost is included. The possible recoverable resources (i.e., material and energy) consist of paper, glass, metal, plastics, steam, and electricity. But these secondary materials could be picked up directly at households or other places rather than in those treatment plants. Thus, a separate term, corresponding to the income from household recycling, is formulated. Since recyclables may not always have economic value in the secondary material market, the plus/minus sign is therefore assigned in the related benefit expressions. The second objective to be maximized is the degree of traffic service at the main entrance road of each treatment or disposal facility. The degree of traffic congestion is conventionally classified as six different levels, each corresponding to a condition of the traffic flow rate relative to the original designed flow rate. The allowable traffic flow is thus equal to the multiplication of the selected service level and the designed flow rate at the main entrance road of each site (Cjkt). ®(Vj.~t) is the average value of background traffic flow rate before the inclusion of the garbage truck fleet, which is recognized as a grey number in this system. The unit used to express Cjkt and ®(Vjkt) is the passenger car unit (P.C.U.). Hence, the traffic impacts created by the operation of solid waste treatment can be expressed by converting the garbage truck fleet into a consistent unit (i.e., P.C.U.) through the use of a conversion factor, CU, associated with the number

E

tcr' kC(J\J~UK\K~)

×CU( (76)

C2(®(Sjkt))

~ @(Sjkt)-~Pl)+@(Vjkt). ic(Ji UKL), ICL (77)

The third objective included is the minimization of noise impacts around each treatment facility. The major sources of noise in a typical solid waste management system include simple source of noise (i.e., from treatment and disposal facilities) and line source of noise (i.e., increased traffic flow by the garbage truck). The former can be properly controlled by engineering technology, but the latter has to be regulated in the optimization process. Although the level of noise, its characteristics, and the criteria used to assess the noise impact, differ from one environment to another, the method of doing so is similar [26, 29-31,40,41,43]. In general, the equivalent noise level (Leq) is the most prevalent approach used for the evaluation of traffic noise impacts. In Taiwan, the degree of noise control in a metropolitan region is classified as four different levels, and the unit used for the description of noise level is dBA. A semi-empirical statistical regression model for noise impact assessment is independently developed by the authors, as illustrated below: NLk = clk + czk in ®m (Fk) -- Dk, ®(Fk) = C U [

~'~

®(Sj~)+PlJ+®(Vjkt)

iE( Ji OKi ), l CL

(78) in which Fk is the noise impact created by the garbage truck fleet at the main entrance road of treatment or disposal facility k. Clk and c2k are regression coefficients. D is the spatial decay constant, an empirical number based on the local situation. The aggregate noise levels, at the most sensible neighboring community around the facility site, can then be estimated and compared with the acceptable noise level required in the environmental regulations. Temporal variations of noise are considered and evaluated through the integration of the noise impacts from those additional

Ni-Bin Chan9 et al./Fuzzy Sets and Systems 89 (1997) 35 60

sources of waste shipping. Therefore, the objective function is:

=

pollutant) at ground level and at centerline of plume may be defined as [14]:

(2)o

C3(®(Sjkt))

Minimize

45

~ ~ ~Clk -I- Czk In kC(J\JIUK\Kt) tET' L

= q @mAkbp

(79) The fourth objective considered is the minimization of air pollution impacts. In Taiwan, air pollution control for municipal incinerators is regulated under the 'Air Pollutants Emission Standards for Waste Incineration' and 'National Ambient Air Quality Standards.' While the criteria for maximum allowable emission rates of pollutants discharged from incinerators are limited by the former, the maximum concentrations (i.e., ppm or p.g/m 3) of certain pollutants in the surrounding environment are controlled by the latter. The objective function is described as below:

(81)

in which Cp(x) is aggregate ambient air pollutant concentration of pollutant p at the downstream location x (gg/m 3 or ppm), u the average wind speed (m/s), He the effective height of plume release corresponding to the wind speed u(m), kp the first order reaction rate of pollutant p (=0 if the pollutant is conserved) (s - l ), t the reaction time (s), q the emission rate of a particular air pollutant from the stack of incinerators (g/s), and az the vertical diffusion coefficient (m). Hence, the FIMOMIP model, based on the idea of weighted additive formulation, is described as below. However, the weight (wi) associated with each fuzzy objective could be fuzzy. In this analysis, the weights are assigned by the analysts or the planners. max

~wi @(/2i)

(82)

i

Minimize

C4(®(Sjkt))

subject to:

I. Fuzzy goal constraint set

kG(K3LJJ3) pGP tGT'

x(

~

(1) goal constraint for cost minimization

@(Sjkt)@(FGR)@(ENp)@(Akbp)) (80)

in which Akbp is the transport and transformation factor that is dependent on the stability, wind speed, distance between emitter and receptor, effective stack height, diffusion coefficient in air, and half life and decay rate of pollutant p [27, 45]. FGR is the flue gas production ratio, based on burning one ton of solid waste in the incinerator. ®(ENp) is the emission factor corresponding to the criteria pollutant p in the flue gas. The multiplication of ®(FGR), ®(ENp), and ®(Akbp) ensures that the more solid waste handled at an incineration site, the greater the amount of air pollution in a designated air quality control region. Such a formulation may yield maximum ground-level ambient concentrations at a set of receptors surrounding the municipal incinerators for air pollution assessment. To determine the whitening value of ®(Akbp), the long-term diffusion equation for a decay-pollutant (non-conservative

G (®(sjk,)) ~< ~ ( f l ) + (1 - @(#l ) ) ( @ ( f l )

--

@(fl ))

(83)

(2) goal constraint for traffic congestion limitation C2( @( Sjkt ) )

~< @(f2) + (1 -- @(k/2))(@(f2) -- @(f2))

(84)

(3) goal constraint for noise impacts control C3( @( Sjkt ))

~< @(f3) + (1 -- @(/~3))(@(f3) -- @(f3))

(85)

(4) goal constraint for air quality control

C4(@(Sjkt)) ~< @_(f4) + (1

-

@(~4))(@(f4)

-

@(f4))

(86)

(5) boundary constraint for membership degree 0~< @(#i)~< 1 Vi

(87)

Ni-Bin Chang et al./Fuzzy Sets and S),stems 89 (1997) 35-60

46

II. Basic Junctional constraint set (1) mass balance constraint: (a) point source: All solid waste generated in the collection district should be shipped to other treatment or disposal components. Furthermore, the waste reduction by household recycling can be taken into account in terms of the participation rate of residents, the recyclable ratio, and the composition of waste. Recycling potential must be evaluated in advance, and the impact on system operations can be shown by including the following constraints.

distortion of the later expansion schedule. T

T

DCk), /> MINk ~ Yky gk E (J2 U J4) )=

1

7"

DCky +

~

NEXPkyt ~< MAXk Ykv

t=y+l Vk E ( J 2 U J 4 ) , V y E ( 1 , T -

gi C ( J I

(s8)

U KI ), VI C T t

";4'(~it) = ~ @(,:~i/,) Vi E (Jl UK]), gt C T' (89) jCR

0 { { ['lt ) { :~ ( ~i[ ..... )

ViE (Jr UK~), V j E R , Vt E T' ,~eO(TR#) =

~'

@(G,,)@(~,,)

Vt e T'

(90) (91)

iC(.]l UKI )

(b) system facility: For any system component, the rate of incoming waste must equal the rate of outgoing waste plus the amount deducted in the treatment process.

NEXPkyl = TEXP~,

( j,k )fill

gk C M, Vt E T'

~

Vk E (J2 U J4), VI C T' (95)

7" N I

T

DCky=MCk ~ ~'~Zkiy Vk E J3 y=¿

(96)

y=l i=I

DCk). +

@(Sik,)

(94)

(b) new facility (incinerators): The numbers of treatment trains and associated size per each combustor in a municipal incinerator should be differentiated in the planning process. Otherwise, the planned size might not be consistent with the industrial specification and reasonable for subsequent engineering design. Hence, the summation of the values of all binary variables Zki), represents the number of combustor being initialized at a specific incinerator site in the time period y. Therefore, Eke.it stands for the choice of expansion in the time period t at an incinerator site, which has been initialized in the time period y.

T

(g'(Sikt)(1- R k ) =

])

t y=2

kG( .l\JI UK\K1 )

(93)

3'= I

~

N2

~ MCk Ekiyt ~ MAXk Yky

t=y+l i=1

(k j)CI2

(92)

(2) capacity limitation constraint: The treatment capacity planned during the procedure of construction and expansion should be less than, or equal to, the maximum allowable capacity and greater than, or equal to, the minimum capacity at one site. (a) new facility (landfills and transfer stations): In the following expression, the binary integer variable is combined with the upper or lower bound of capacity such that the site selection can be performed by the binary choice of its value 'one or zero', which corresponds to the 'inclusion or exclusion' of design capacities in the constraint and related cost/benefit terms in the objective function. The period of facility initialization is denoted by the symbol 'y' that can avoid

Vk c J3, Vy E (1, r -

1)

N2 ~ M C k Ekyit = TEXPk, .},=2 i=1

(97)

t

VkEJ3, V t E T '

(98)

N 1

}-~ Zk,y ~> Yk), Vk E J3, Vy E ( 1 , T - 1)

(99)

i=1

(c) old facility: T

DCk + ~ TEXPk, ~ MAXk

Vk E (K\K~)

(100)

t=l

(3) operating constraint: the accumulated waste inflow at each site should be less than, or equal to, the available capacity in each planning period.

Ni-Bin Chang et aL /Fuzz), Sets and Systems 89 (1997) 35 60 (a) new facility: TIME

garbage truck stream.

DCky + L y= 1

>>- ~ ~(sik,) (j,k)El~

~" NEXPkyt t=),+ 1

)l

Vk c (J\J1), Vt' E T'

(101)

,

TIME DCk + ~ TEXPkt

~

~(sik,)

)

vk ~ (K\K1), Vt' ~ r ' (102)

(./,k )Cl~

T

Ykt <~ ]

c,k + c2k In~CU [ ~ L LiE(JIUKt),IEL

Vk ~ (J2 U J3 U J4)

(106)

®(Sykt)+P,]

+ Q (V~-kt)}~< NLk Vk, Vt C r '

(4) conditionality constraint: the conditional constraint ensures that the initialization of a new site in a system can only occur once in a multistage planning project.

t

(vj~,) ~< SLjk, cjk,

(2) noise control constraint:

t=l

>/

+ o

Vj E (J\J1UK\K1), Vt C T'

(b) old facility:

(

47

(103)

(107)

(3) air pollution control constraint: This analysis considers ambient air quality limitations for several pollutants at a set of prespecified sensitive areas in Kaohsiung City. The constraints formulation are described as below:

1

(5) site availability constraint: this constraint can also allow the planner to leave out some of the potential sites.

f' [

~

~

(~(Sj~t) @(FGR) ~(ENp)

L kC(K3UJ~ ) (j,/,-)EI~

@(Akbp))[ <. Spt

@Bbpt Vb, Vp E P, Vt E T I

T

Ykt~
Vt ~ r '

(]08)

(104)

y- 1

(6) financial constraint: The key point in the formulation is the use of an inequality rather than equality, constraint. If the equality constraint holds, the solution will show that there will never be profits in operating these facilities in each period, and the accumulated income will be used up through the building of extra treatment capacity which is of no use in that period. @(Ct) <~ O(Bt) + TIPt [ic(K~j,) ~(Git)l

Vt C T t

(lO5) Ill. Environmental quality constraint set (1) traffic congestion constraint: SLjkt represents the selected service level of traffic flow at each facility sites. The allowable traffic flow is thus equal to the multiplication of the selected service level and the designed flow rate at the main entrance road of each site (Cjkt), as shown on the right-hand side of the constraint below. GVjkt is the average value of background traffic flow rate before the inclusion of the

f t is a conversion factor regarding the time scale difference between the units of emission rate and National Ambient Air Quality Standards (Spt). The variable ~)Bbptin the right-hand side of the constraint serves as an input variable to show the background concentration of air pollutant ' p ' at the location of a specific receptor 'b' at time t.

5. Case study

5.1. System environment of Kaohsiung metropolitan region Kaohsiung City, located beside Kaohsiung harbor, is the largest city in the southern part of Taiwan. The geographical location of this city and the proposed solid waste management system are shown in Fig. 1. Twelve garbage collection teams are in charge of the clean up work in the eleven administrative districts. Only the Sanming district owns two collection teams, and the service area is separated by east

Ni-Bin Chang et al. / Fuzzy Sets and Systems 89 (1997) 35-60

48

Table 1 Construction schedule and design capacity of three incinerators Location

Nantzu Fuhdingjin Talinpu

Construction schedule (start up year)

Design capacity (tons/day :TPD)

Phase I

Phase II

Phase I

Phase II

1999 1996 1998

2003

1200 900 1800

1200

2001

Status

1200

EIA EIA, planning EIA, planning

Source: EPB, Kaohsiung City Government. and west divisions. The only existing landfill is the Shichinpu landfill, located at the northern boundary of Kaohsiung. In addition, there is an existing transfer station in the Chichin district, which is a separate island on the other side of Kaohsiung harbor. The transportation to Chichin mainly relies on an underground tunnel across the bottom of the harbor connecting with the downtown area of Kaohsiung City. Three sample sites - Fuhdingjin, Nantzu, and Talinpu are planned for future resource recovery plants. Two proposed sites of transfer stations (Tsoying and Chienchen) and one new landfill (Tapindin) were selected in a preliminary screening procedure. But uncertainties still exist in the procurement o f the land and the agreement of local residents. The Shichinpu landfill is expected to be closed in 1995, but it has to be expanded in the near future due to the lack of other treatment alternatives in the current solid waste management system. Table 1 shows the proposed solid waste management program of Kaohsiung City in the future. Several key questions frequently bother the public officials, which include: (a) Is it necessary to build two new transfer stations? (b) Are the construction schedule and planned capacity reasonable to meet the growing demand of waste treatment? (c) What is the impact of material recycling on the entire management system? and (d) What is the long-term optimal waste management pattern once the environmental quality considerations are included in the next twenty years? These questions can be analyzed using this multiobjective programming model. 5.2. Analytical f r a m e w o r k

In this analysis, a hypothetical 20-year project with four time periods is conducted. The start-up year is 1994, when the system has only one landfill and one

transfer station. The Shichinpu landfill is expected to be expanded and continuously used until the year 2004 (i.e., the end of second time period). The start-up date o f operation of the Tapindin landfill is assumed to be at the beginning of the second time period. The Chichin transfer station, which only serves the Chichin district, is regarded as a point source. Construction or expansion of any facility is to be completed within the previous time period. If a facility is to be used in time period t, then it must be constructed in time period t - 1 or before. Hence, the use of any facility in the dynamic optimization process represents the start-up date o f its operation, whenever investments are incurred. Therefore, the potential sites o f transfer stations and incinerators can be included into the system operation after the beginning o f the second time period. The candidate sites for transfer stations are prepared for shipping raw garbage only. 5.3. Data acquisition and analysis

A lot o f physical, economic, and environmental data for solid waste management have to be compiled together to build up the objective functions and constraints. Especially, various investigations for those supporting submodels in environmental quality constraints have to be conducted before the optimization procedure is performed. Economic database is mainly collected from the government agencies. Final selections of each parameter value need to be reviewed by many disciplines. After a series of investigations, an independent regression analysis for the determination of the fixed and variable costs in the construction cost functions is applied, based on a local database of landfills and incinerators. Since there were no formal transfer stations

49

Ni-Bin Chan9 et al./ Fuzzy Sets and Systems 89 (1997) 35-60 Table 2 Construction cost and design capacity of existing municipal incinerators in Taiwan Sites Neihu Mucha Shulin Hsintien Taichung City Chiayi City Homei Tainan city CKK airport III CKK airport Kaohsiung airport

Bidding year

Capacity (tons/day)

General index in construction"

Construction cost ( 104NT $ )

Adjusted construction cost ( 104NT $ )b

1987 1990 1988 1988 1992 1993 1989 1993 t991 1990 1988

3x 4 x 3x 2× 3× 2× 30 2x 20 3 3

104.84 133.27 113.07 113.07 158.15 168.43 127.52 168.43 138.61 133.27 113.07

160 000 327 000 492 750 328 500 325 000 163 004.6 11 780 276478.1 14500 5 700 5 398

257 046 413 271 734004 489 336 346 125 163 004.6 15 559 276478.1 17619 7 203 8 040

300 375 450 450 300 150 450

aGeneral index in the 'Commodity Price Statistics Yearbook: Taiwan Province, 1991'; the basis is 100 in 1986. bAdjusted to the year 1993 by construction cost index. Table 3 Construction cost and design capacity of existing transfer stations

Sites

Design capacity (tons/day)

Recorded construction cost (NT $)

Bidding or recorded year

Calibrated construction cost (NT $)a

site 1 site 2 site 3 s~te 4 site 5 site 6 site 7 site 8 site 9 site 10 site 11

387 120 289 112 190 600 228 t 55 220 200 830

7 040 000 22 320 000 3 076 000 10 960 000 11 640 000 31 920 000 34 400 000 5 240 000 9 760 000 7 200 000 57 560 000

1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975

16 966 400 53 791 200 7 413 160 26 413 600 28 052 400 76 927 200 82 904 000 12 628 400 23 521 600 17 352 000 138 719 600

aSite 1-site 11 were provided by Booz-Allen and Hamilton Inc., (1976) and are calibrated to the year 1993 by the construction cost index in ENR

in T a i w a n , U S data w e r e u s e d after a careful calibration [2]. Table 2 - 4 list the i n f o r m a t i o n r e q u i r e d for the linear r e g r e s s i o n analysis o f c o n s t r u c t i o n c o s t functions c o r r e s p o n d i n g to incinerator, t r a n s f e r station, and landfill, r e s p e c t i v e l y . Facility e x p a n s i o n costs are a s s u m e d to b e the s a m e as the variable costs in t h e s e c o n s t r u c t i o n c o s t f u n c t i o n s . In addition, the prices o f electricity and s e c o n d a r y materials, the interest rate a n d inflation rate, o p e r a t i n g c o s t o f the t r e a t m e n t and d i s p o s a l facilities, and the t r a n s p o r t a t i o n c o s t w e r e s e p a r a t e l y i n v e s t i g a t e d or e s t i m a t e d as well. W a s t e

r e d u c t i o n ratio and the c o n v e r s i o n efficiency o f energy r e c o v e r y c o r r e s p o n d i n g to the i n c i n e r a t i o n proc e s s w e r e s e l e c t e d a c c o r d i n g to several e n g i n e e r i n g reports. The results o f l i n e a r - r e g r e s s i o n are: (for incinerators) TC =

144591961 (0.288)

R 2 = 0.8089

+ 3774074 (6.173)

× DC

50

N#Bin Chan q et al./ Fuzz), Sets and Systems 89 (1997) 35-60

Table 4 Construction cost and design capacity of existing municipal landfills (valley type) in Taiwan

Sites

Bidding year

Capacity (tons/d)

General index in constructiona

Construction cost (NT $)

Adjusted construction cost (NT S) b

Mucha Taoyuan Keelung Sanhsia Pall Yungho Pinglin Tungkong Fangliao Hengchun Joulu

1983 1987 1990 1991 1992 1993 1992 1989 1990 1988 1987

2822 353 540 200 100 240 20 50 70 50 20

99.9 100.84 133.27 138.61 158.15 168.43 158.15 127.52 133.27 113.07 104.84

674 547 000 151 668 000 664 000 000 259 800 000 28 890 000 43 200 000 23 800 000 97 563 250 132 488 624 c 115851659 59 104919

1 139 984 430 254 802 240 839 180 010 315 692 33 I 30 767 895 43 200 000 25 347 037 128 862 752 167 442 477 172573582 94954612

aGeneral index in the "Commodity Price Statistics Yearbook: Taiwan Province, 1991"; the basis is 100 in 1986. bAcljusted to the year 1993 by construction cost index. Cphase I only. Table 5 Waste generation record in Kaohsiung City (tons/day) Year No.

1989

1990

1991

1992

1993

Average increasing rate (%)

87.6 116 94.1 318.9 124.0 194.9 44.6 55.7 97.9 215.0 188.9 37.3 78.9 1334.9

103.7 134.5 108.0 356.3 205.8 150.5 47.6 60.5 110.4 238.5 221.2 44.3 80.4 1522.2

121.5 158.9 116.1 389.5 226.0 163.5 51.9 69.9 122.7 265.5 294.9 41.8 96.5 1729.2

16.3 15.7 9.7 10.8

District

1 2 3

Nantzu Tsoying Kushan Sanming East side West side Yencheng Chienchin Hsinhsing Linya Chienchen Chichin tlsiaokang Total

4 5 6 7 8 9 10 II 12

66.6 88.8 80.1 258.4

81.0 102.3 87.7 295.0

42.1 50.5 87.5 183.9 153.5 22.7 65.4 1099.5

44.7 53.9 94.4 202.7 165.2 27.2 77.1 1231.2

5.4 8.5 8.9 9.6 18.1 17.5 10.5 11.9

Source: EPB, Kaohsiung City Government. (for transfer stations) T C -- 3 2 7 4 0 8 1 (0.219)

+

134697

T h e v a l u e s in t h e a b o v e p a r e n t h e s e s a r e t - r a t i o s w h i c h × DC

(3.342)

good. The historical records of solid waste generation w e r e c o l l e c t e d , a s l i s t e d in T a b l e 5, a n d w a s t e g e n e r a -

R 2 = 0.5538

(for landfills) TC -- 136540517 (2.081) R2

0.7454

a r e s t a t i s t i c a l l y s i g n i f i c a n t in m o s t c a s e s u n d e r t h e 5 % l e v e l o f s i g n i f i c a n c e . T h e R 2 v a l u e s a r e all r e a s o n a b l y

+ 3 8 3 172 x D C (5.133)

tion rates were forecasted by linear regression model c o r r e s p o n d i n g to e a c h d i s t r i c t . T h e 9 5 % c o n f i d e n c e interval

is s e l e c t e d

for the determination

of grey

Ni-Bin Chang et al./ Fuzzy Sets and Systems 89 (1997) 35 60

5

Table 6 The prediction of waste generation by grey numbers Administrative districts

Year

Nantzu Ysoying Kusban Sanming east Sanming west Yencheng Chienchin Hsinhsing Linya Chienchen Chichin Hsiaokang

[180.59, [235.65, [159.19, [316.66, [229.10, [60.29, [86.75, [158.88, [354.43, [416.15, [68.74, [119.86,

1999

2004 189.07] 245.91] 164.43] 321.68] 232.70] 63.57] 93.01] 167.25] 366.42] 467.69] 78.00] 131.16]

2009

[244.28, [318.77, [203.75, [409.03, [295.91, [70.57, [107.58, [199.59, [450.33, [570.08, [93.63, [149.21,

257.88] 335.19] 212.17] 417.02] 301.69] 75.79] 117.58] 212.94] 469.51] 652.52] 108.41] 167.31]

2014

[307.89, [401.78, [248.26, [501.32, [362.69, [80.81, [128.33, [240.19, [546.10, [723.44, [118.40, [178.44,

326.77] 424.58] 259.96] 512.42] 370.72] 88.05] 142.23] 258.73] 572.74] 837.96] 138.94] 203.58]

[370.84, 395.33] [484.75, 514.02] [292.76, 307.77] [593.61, 607.851 [429.45, 439.75] [91.03, 100.33] [149.06, 166.90] [280.76, 304.561 [641.83, 676.02] [876.60, 1023.6] [143.14, 169.50] [207.63, 239.90]

Unit: tons/day. Table 7 Physical composition of municipal solid waste in Kaohsiung city Year

1985

1986

1987

1988

1989

1990

Combustible Paper Textiles Trimmings Food waste Plastics Rubber Others

85.35% 19.66% 5.96% 10.07% 20.05% 18.41% 0.1% 10.65%

84.14% 19.25% 7.91% 8.15% 24.07% 15.52% 0.74% 8.5 %

---------

83.68% 19.70% 4.2 % 3.00% 35.15% 18.06% 2.23% 1.34%

83.32% 22.00% 3.1% 5.96% 31.54% 16.89% 1.77% 2.06%

83.34% 2 1.98% 3.56% 5.84% 29.06% 19.15% 2.18% 2.08%

Noncombustible Metal Glass Ceramics Stone, sand Others

14.65% 1.79% -3.42% 9.44%

15.86% 2.78%

---

4.75% 8.33% --

16.32% 7.17% 5.47% 1.38% 2.30%

16.38% 7.93% 6.03% 0.38% 1.64%

--

16.68% 7.40% 5.41% 1.55% 2.32% --

Item

Source: 1991 Yearbook of Environmental Statistics: Taiwan area, R.O.C. On dry basis. Table 8 The grey numbers of the upper bounds of recyclables in the waste stream 1999 Paper Plastic Glass Metal Unit: %.

[0.13, [0.10, [0.01, [0.03,

2004 0.20] 0.15] 0.05] 0.101

[0.13, [0.10, [0.01, [0.03,

2009 0.20] 0.15] 0.05] 0.10]

[0.13, [0.10, [0.01, [0.03,

2014 0.20] 0.15] 0.05] 0.101

[0.13, [0.10, [0.01, [0.03,

0.20] 0.15] 0.05] 0.10]

Ni-Bin Chang et al./Fuzzy Sets and Systems 89 (1997) 35-60

52

Table 9 Classification of traffic service level of different types of road Service level classification

Explanation of service level condition

Average speed (km/h)

/~/C

A B C D E F

Free traffic flow Steady traffic flow (slightly delay) Steady traffic flow (acceptable delay) Close to unsteady traffic flow Unsteady traffic flow Traffic jam

~>50 ~>40 />30 ~>25 >~25 --

~<0.6 ~<0.7 ~<0.8 ~<0.9 ~< 1.0 --

Table 10 Design capacity of fast lane corresponding to road pattern Class of road

Class no.

Pattern coefficient

Number of fast lane

Design capacity of fast lane

Free way

1

1.8

n

1000 x n x 1.8

Express way

2

1.5

n

1000 x n x 1.5

General road: safety island and separation of lane separation of lane marking only safety island standard safety island no lane marking

3 4 5 6 7

1.3 1.1 1.0 0.8 0.6

n n n n n

1000 1000 1000 1000 1000

-

-

-

x x x x x

n n n n n

x x x x x

1.3 1.1 1.0 0.8 0.6

Unit: P.C.U. P.C.U. = (big truck or big bus) x 2 + (small truck or small bus) x 1 + motorcycle x 0.5 + trailer x 3

Table 11 The investigation of traffic service level of different roads at different facilities

Facility name

Main entrance road

Class no.

No. of fast lane

Average traffic flow (P.C.U./h)

Designed traffic flow (P.C.U./h)

/~/C"

Service level

Ho-Chun Rd. Ming-Ju Rd. Yen-Hai II Rd.

4 3 4

4 4 4

[850, 950] [3100, 3300] [3050, 3200]

1600 5200 4400

[0.53, 0.59] [0.60, 0.63] [0.69, 0.73]

C C C

Boy-Ay Rd. Jong-Shan 1II Rd

3 4

4 6

[1500, 1650] [4520, 4760]

5200 6600

[0.29, 0.32] [0.68, 0.72]

Gao-Nan Rd. Kao-Fun Rd.

4 6

4 4

[3450, 3700] [1100, 1200]

4400 3200

[0.78, 0.84] [0.34, 0.38]

Incinerator: Nantzu Fuhdingjin Talinpu

Trans[er station: - Tsoying - Chichin

B

C

Landfill: Shichinpul -- Tapindin

D A

53

Ni-Bin Chan 9 et al./ Fuzzy Sets and Systems 89 (1997) 35-60

Table 12 Investigation of noise levels and related data at different facilities

Name of each facility

Name of main entrance road

Distance between the facility and nearest community (km)

Distance between the entrance road and nearest community (km)

Required noise control levels in the nearest community (dBA)

Background noise level of traffic stream (dBA)

Jiun-Shiaw Rd. Ming-Ju Rd. Yen-Hai IV Rd.

2 0.3 0.3

0.0 0.3 0.3

65(II)a 60(I) 60(1)

69.5 62.9 74.3

Jong-Jou II Rd. Boy-Ay Rd. Jong-Shan IIl Rd.

0.0 0.2

0.0 0.2

75(IV) 65(II)

71.2 78.6

Gau-Nan Rd. Fei-Ji Rd.

0.5 0.0

0.1 0.0

60(1) 65(II)

72.5 65

Incinerator:

Nantzu Fuhdingjin Talinpu

-

-

-

Transfer station:

- Chichin Tsoying - Chienchen Landfill: -

-

Shichinpul Tapindin

a The roman numerals in the parentheses are the class of noise control determined by both the required upper bound of noise level close to the road the noise control limitation in R.O.C.

Table 13 The Investigation of air quality impacts by incineration projects

Monitoring location Distance between emitter and receptor (m) Wind speed (m/s) Back ground concentration: TSP (mg/m 3) SO2 (ppm) Effective stack height (m) Dispersion parameter 6z SO2 decay rate SO2 transfer coefficient TSP transfer coefficient

Nantzu, Yen-Jun Primary School

Fuhdingjin, Din-Jin Junior High School

Talinpu, Chian Steel Inc.

1400 3.5

1500 2.8

1200 2.5

[80, 105] [7, l l] 130 40.02 0.98 4.54 × 10 -6 4.64 x 10 -6

[170, 210] [10, 15] 131 41.96 0.97 7.02 × 10-6 8.28 x 10-6

[165, 200] [30, 40] 101 35.93 0.98 2.67 × 10-5 2.72 x 10-5

n u m b e r s o f w a s t e g e n e r a t i o n , as s h o w n in T a b l e 6. A c c o r d i n g to t h e 5-yr r e c o r d o f p h y s i c a l c o m p o s i t i o n o f solid w a s t e g e n e r a t e d in K a o h s i u n g city, as s h o w n in T a b l e 7, ~ijt, max o f paper, plastics, m e t a l a n d glass c a n b e d e t e r m i n e d r e s p e c t i v e l y , as listed in T a b l e 8. T h e a v e r a g e t r a n s p o r t a t i o n d i s t a n c e a m o n g different s y s t e m c o m p o n e n t s w e r e e s t i m a t e d to f u r t h e r prov i d e the t r a d e - o f f b a s i s d u r i n g o p t i m i z a t i o n p r o c e s s . The maximum and minimum capacities of t h o s e t r e a t m e n t a n d d i s p o s a l sites w e r e d e c i d e d

a c c o r d i n g to the l a n d a v a i l a b i l i t y a n d t e c h n o l o g y information. In a d d i t i o n , the c l a s s i f i c a t i o n o f traffic s e r v i c e level a n d d e s i g n c a p a c i t y o f fast lane c o r r e s p o n d i n g to r o a d p a t t e r n w e r e c o l l e c t e d a n d i l l u s t r a t e d in T a b l e s 9 a n d 10. B a c k g r o u n d traffic flow at t h e m a i n e n t r a n c e r o a d at e a c h site, a n d r e l a t e d n o i s e i m p a c t s w e r e m e a s u r e d a n d s u m m a r i z e d , as s h o w n in T a b l e s 11 a n d 12. T h e g r o w t h rates o f b a c k g r o u n d traffic flow o v e r t i m e p e r i o d s c a n b e a s s u m e d to b e at the s a m e s p e e d as

54

Ni-Bin Chang et aL /Fuzzy Sets and Systems 89 (1997) 35 60

0

l

l

I if Cl(®Sjkt) < 38876 (629108 - Cl(®Sjkt) ) ff 38876 < Cl(®Sjkt) < 629108 if Cl(®Sjkt) > 6291(18

38876

629108

:" Cl(®Sjkt)

C o s t ( 1 9 9 3 m i l l i o n s NT$)

(1.1409 - ~z (®Sjkt))

if C2(®Sjkt) <~ 0.4523 if 0.4523 < C2(®Sjkt) _< 1.1409 if C2(®Sjkt) > 1.1409

0

04523

1. 1409

C2( ® Si~)

Traffic congestion index

?oT

®,u3 = (31.36 - C3(®Sjk t)) 2656 0

if C3(®Sj~ ) _< 4.80 if 4.80 < C3(®Sjk t) _< 31.36

if C3(®Sjkt)> 31.36

C3( ® Sjkd

0

480

31.36

Noise impacts (dBA)

(820.89 - ~a (®Sjla)) ®'Lt4 = t

0

~-

if C4(®Sj!~) < 14.75 if 14.75 < C4(®Sj~) _< 820.89 if C4(®Sjk0 > 820.89

) C4(~) Sjkt) 14.75

820.89

Air quality (mg/m 3) Fig. 3. The determination of membership functions in FIMOMIP model.

55

Ni-Bin Chan 9 et aZ /Fuzzy Sets and Systems 89 (1997) 35-60

Table 14 Optimal results in system analysis Objective function value ( 1993 millions NT$)

[ 143 300, 164 602]

Degree Degree Degree Degree

[0.787, [0.598, [0.826, [0.731,

of of of of

membership membership membership membership

function function function function

of of of of

cost minimization traffic congestion limitation noise impacts control air quality control

New incinerator sites included Time period 1I Time period II Design capacity (TPD) Time period II Time period II Expansion capacity of Nantzu plant (TPD) Time period Ill Time period IV Expansion capacity of Fudingjin plant (TPD) Time period llI Time period IV Expansion capacity of Talinpu plant (TPD) Time period Ill Time period IV New transfer station included design capacity (TPD) expansion capacity of Tsoying (TPD) expansion capacity of Chienchin station (TPD) New landfill included Time period 11 Design capacity (TPD) Time period II Expansion capacity (TPD) Time period III Time period IV Tipping Time Time Time Time

fee (NT$/ton) period I period II period Ill period 1V

Recycling (TPD) Time period l Time period II Time period Ill Time period IV

0.823] 0.614] 0.876] 0.758]

Nantzu Fudingjin 1 x 450 1 x 450 1 × 450 1 x 450 0 0 m

m

m

Tapindin 3000

m

[2254.4, 2401.8] [2379.1, 2467.7] [866.5, 1782.8] [12529.6, 14068.7]

[0, o] [0, o] [0, 47.9] [743.8, 856.8]

56

Ni-Bin Chan9 et al./Fuzzy Sets and Systems 89 (1997) 35 60

Nantzu

1

Nantzu I[.-.

Shichinpul

~2

Tsoying

@

~3

Kushan

[307.9.322.9] (111) [310.6,325.91 (IV) - - - - . . . . . ..

130.6,47.01 (11) . . . . . '1401.8,419.6] (III)

i

Tapindin

Tsoying

'.

•, ,

~ 1

"[59.6,59.61 (1I)

[228.E22~61(II)

J •.

'"l

~i"x

[107.6,117.6] (II) ] [12R3,I40.6] (I11) [124.5.138.61 (IV) [199.6.212.91 (II) I240.2255.71 (lid [234.6,251 4] (IV)

i'° •

¢7

Chienchin

~8

Hsinhsing

¢9

Linya

~10

Chienchen

Oil

Chachin

[33.4,36.4] (II) [48.1,49.2] (III) "~49.2,49.2] (IV)

".

[409.0,417.0] (11) [188.9,188.9] (III) [182.4,182.4] (IV)

"(

. •",

~i,,.i

""',

•,.

"" ~

'

""''• '~'"'" '

-

~

-,

•

"""

""" -""~'::-[" -

[295.9.301.7] (II) [362.7.366.3](III) [361.8,361.8](IV)

'

"'. • •.

'

Yencheng

I312.4,317.5] (III) [317.5,317.5] (IV)

~4

J'.

¢6

~

~5 ""

[70.6.75.8] (II)

Sarurung east Sanrmng west

~ 1 2 Hsiaokang

"-..

1141.0.149.7] (lIl) :1134 3.142 2] (IV)

~3 : 3.8,212.2iiIr).. I

(II)

¢4 ~5

~

"" "'

[450.3 469.5] (II) [

"'"~-. ,

[546.1,566.0] (lid ,. [538 4,556.9] (IV)I

¢ 10. f~570.1,65")'J$~'('hl)i'"'~-~, .... ..... ~.,

~..

Legend ]1~

Sources of waste generation

•

E:,astmg treatment facilities

•

Proposed sites of resource recovery.

"'::. " "~" q-72.3.4.838.0] ira) " "- ":'-. ", ", [799~4,9"2-3.~](IV)"" :: : ' , ' - : ~

~ ~:Tapldin

] [93 6.108.4i(1"~) . . . . . . ". . . . . " . " ~ 1118.4,1389] (IlI) ......... ".~ 1130.4,153.01 (IV)

Proposed site of sanitary, landfill

l

]

Talinpu• ~ 12/ [ 149.2,167.3~I) [ 17g.4.20,1'/31 (III)

Proposed sites of transfer stations * The Roman numerals in the parentheses represent the time period in each case ** unit tons/day Fig. 4. Optimal waste flow pattern of the case study (unit: tonnes/day).

Ni-Bin Chang et al./ Fuzzy Sets and Systems 89 (1997) 35-60

the estimated economic growth rate. Background information of criteria air pollutants and related meteorological conditions at each designated site were investigated or obtained from several reports of environmental impacts assessment, as listed in Table 13. 5.4. Results and discussion

To determine the fuzzy membership functions, a pay-off table is required to estimate the tolerance intervals of objectives in decision making. Such an analytical scheme is frequently applied when the system configuration is too complicated to be handled by the decision makers directly. Fig. 3 shows the fuzzy membership function applied for each objective. The inclusion of a weight associated with each fuzzy objective may provide an additional flexibility in decision making. Such an arrangement implies that environmental considerations are favored if each objective is assigned by the same weight. However, to avoid exaggerating the importance of environmental objectives, equal weight is assigned to each group of economic and environmental objectives in this case study. The fuzzy interval optimal outputs are then obtained and listed in Table 14. Hence, the optimization results unambiguously indicate that the inclusion of environmental quality objectives would rule out the possibility of Talinpu incineration project, which is close to the coastal industrial complex, due to the noise control and traffic congestion limitations. Such an option would indirectly increase the need of landfilling space, while material recycling is not favored due to the rising recycling cost. It is worthwhile to point out that both the proposed sites of transfer stations are excluded in the fuzzy multiobjective analytical framework. This is due to the fact that the noise and traffic impacts cannot be tolerated at the proposed sites of these two transfer stations, based on the current environmental regulations. Once the waste stream is mainly allocated onto the landfill, the tipping fees may dramatically increase over periods. Fig. 4 delineates the optimal pattern of waste stream allocation over planning time periods.

57

a set of more flexible optimal solutions in solving complicated real-world issues. The fuzzy interval outputs can successfully integrate system complexity regarding social, physical, economic, environmental, and institutional uncertain messages to generate more acceptable and adjustable management policies. The case study is prepared not only for achieving the long-term economic planning, but also for considering the operational impacts on environmental quality in an integrated solid waste management system. In the area of sustainable development for metropolitan solid waste management, such a system analysis with uncertainty is concemed with all aspects of the economic efficiency, recycling requirements, and environmental impacts for those communities affected by the facilities siting and operation in the selected solid waste management altematives. It is believed that the FIMOMIP approach represents a significant improvement in both the theory of multiobjective programming and the application for long-term planning of solid waste management systems. Appendix. Notation used in the model Definition o f sets

I 1l 12 J J1 J2 J3 J4 K

6. Conclusions

Kl

This analysis has shown that the proposed FIMOMIP model is an effective tool for generating

K2

set of linkages between system components in the transportation network in each period set of incoming waste stream at a specific site in each period set of outgoing waste stream at a specific site in each period set of all new system components (J1 U J2 U J3 U J4) in each period set of all new waste generation districts (point sources) in the system set of all new waste transfer stations in the system in each period set of all new waste treatment plants in the system in each period set of all new waste landfills in the system in each period set of all old system components (K1 U K2 U K3 U K4) in each period set of all old waste generation districts (point sources) in the system in each period set of all old waste transfer stations in the system in each period

Ni-Bin Chang et al./Fuzzy Sets and Systems 89 (1997) 35 60

58

/(3 K4 L R T' M

set of all old waste treatment plants in the system in each period set of all old waste landfills in the system in each period set of types of trucks used for shipping waste in the system set of resources recovered at facilities and households set of time period ({ 1. . . . . T}) set of all intermediate facilities in each period

Definition of input var&bles Akbp

$¢ijt, max

Bbpt

CCkt COkt CRjt CTjk, CU

Cjkt

Dj f f'

Fkt FGR Git

the transport and transformation factor corresponding to the linkage between plant ' k ' and receptor 'b' for pollutant 'p' maximum fraction of recyclables which can be recovered in the waste stream Git the background concentration of air pollutant p at the location of a specific receptor a at the time period t discount factor for time period t variable construction cost at facility k at time period t unit operating cost at facility k at time period t recycling cost of material i at time period t unit transportation cost among system components at time period t the conversion factor between the garbage truck unit and passenger car unit the maximum designed traffic capacity on the main entrance road at each facility at time period t spatial decay constant at site j, based on the local situation estimated inflation rate a conversion factor regarding the time scale difference between the units of emission factor (ENp) and National Ambient Air Quality Standard (Spt) fixed cost for building new facility at site k at time period t the flue gas production ratio, based on burning one ton of solid waste in the incinerator waste generation rate in municipal district i at time t

IRijt

net income per unit weight of secondary material j by household recycling in district i and at time period t LIMITk total tolerance of pollutant p in the incoming waste stream at landfill k MAXk the maximum allowable capacity at site k MINk the minimum required capacity at site k MCk the design capacity of a basic unit of a combustor in each treatment train at site k, which is consistent with the industrial specification N1 The maximum number of treatment train for waste incineration which can be initialized in the optimization process N2 The maximum number of treatment train for waste incineration which can be expanded in the optimization process N~ the specified number of available potential sites in a time period N~ the acceptable noise level of site j in the environmental regulations PI the allowable weight loading of different types of trucks the price of each resource i recovered at site Pikt k at time period t F nominal interest rate Rk reduction ratio of waste destroyed by the processing at site k and time t the emission standard of pollutant p at the Spt time period t SLjkt required service level of main road connecting different system components at time period t T the number of total time periods in the planning horizon recovery factor of resource i per unit waste Tikt processed at facility k at time period t TIME the length of time within one time period t (conversion factor) the average background traffic flow on the Vjkt main entrance road at each facility at time period t the weight associated with each objective asWi signed by decision makers

Definition of decision variables ~t

total recycling fraction corresponding to waste inflow Git

Ni-Bin Chang et aL /Fuzzy Sets and Systems 89 (1997) 35-60

Ct,Bt

c1,c2,c3,c4 DCk, Ekiyt

fl,f2,f3,f4

]/i NEXPkyt

Sjkt

TEXPkt TIPt TRt

rk~ Zkiy

r e c y c l i n g fraction o f material j corres p o n d i n g to waste Git the total system costs and benefits respectively at t i m e period t the f u z z y m e m b e r s h i p function values c o r r e s p o n d i n g to these four objectives design capacity o f a n e w facility at site k at t i m e p e r i o d t binary integer variable for the selection the ith treatment train o f facility k in the e x p a n s i o n time period t, initialized in the t i m e period y the possible o b j e c t i v e function values, prepared for the construction o f f u z z y m e m b e r s h i p functions for those objectives f u z z y m e m b e r s h i p degree for the ith objective expansion capacity at n e w site k at time t based on the initialization o f facility operation at time period y optimal waste stream a m o n g system c o m p o n e n t s at time period t total expansion capacity o f a n e w or an old facility at site k at t i m e t tipping fee charged per unit a m o u n t o f waste at t i m e period t total a m o u n t o f h o u s e h o l d r e c y c l i n g at time period t binary integer variable for the selection o f facility at t i m e period t binary integer variable for selection o f the ith treatment train o f facility k in the initialization t i m e period y

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59

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